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Sheaves over finite DAGs may be archetypal
(Or: ``Sheaves for non-categorists''. \red{Work in progress})
\msk
Eduardo Ochs - PURO/UFF
eduardoochs@gmail.com
\url{http://angg.twu.net/}
\url{http://angg.twu.net/math-b.html#sheaves-on-zdags}
\msk
Presented at the XVI EBL
(\url{http://www.cle.unicamp.br/ebl2011/}),
held at Petrópolis, RJ, Brazil, on 2011may13.
\red{These slides will probably be updated soon}
\red{to make them more self-contained.}
\newpage
%*
% (eedn4-51-bounded)
Index of the slides:
\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")
\tocline {Let's mystify the audience with technical terms} {3}
\tocline {Let's mystify the audience a bit more} {4}
\tocline {Well-positioned subsets of $Z^2$ (and ZSets)} {5}
\tocline {Black pawn's moves (and ZDags)} {6}
\tocline {Partial orders} {7}
\tocline {Cycles are evil} {8}
\tocline {DAGs are good} {9}
\tocline {Our favorite topological space: ${\mathbb {V}}$} {10}
\tocline {Our favorite sheaves and presheaves} {11}
\tocline {Compatibility} {12}
\tocline {The evil presheaf} {13}
\tocline {Stack operations} {14}
\tocline {Stack operations (2)} {15}
\tocline {Covers and families} {16}
\tocline {Saturated families} {17}
\tocline {Adding unions} {18}
\tocline {Priming} {19}
\tocline {What next?} {20}
\newpage
% --------------------
% «mention-modal» (to ".mention-modal")
% (s "Let's mystify the audience with technical terms" "mention-modal")
\myslide {Let's mystify the audience with technical terms} {mention-modal}
\msk
{\bf Modal logic:}
\par S4 has the finite model property.
\par We have Gödel's translation: intuitionistic logic $\to$ S4
\par So: as $¬¬P ⊃ P$ is not a theorem of intutionistic logic
\par $\funto$ there is a finite model (with two worlds)
\par \lph{$\funto$} in which $¬¬P⊃P$ is not true.
\par These finite counter-models are good for developing
\par intuition about intuitionistic logic.
\msk
\par {\bf Category Theory:}
\par Let $\bbW$ be a finite poset.
\par ($\bbW$ is a system of possible worlds for S4,
\par viewed as a category).
\par Then $\Set^\bbW$ is a topos of presheaves.
\par The logic of toposes is intuitionistic,
\par and in $\Set^\bbW = \Set^{*\to*}$ we can falsify $¬¬P⊃P$.
\msk
\par Claim:
\par Toposes of the form $\Set^\bbW$ are good for developing
\par intuition about Topos Theory (and CT in general).
% --------------------
% «mention-sheaves» (to ".mention-sheaves")
% (s "Let's mystify the audience a bit more" "mention-sheaves")
\myslide {Let's mystify the audience a bit more} {mention-sheaves}
\msk
\par Sheaves are very important in Topos Theory.
\par Category Theory is \red{hard} (too abstract).
\par Even \red{basic sheaf theory} is \red{too hard}.
\par Idea: Let's use toposes of the form $\Set^\bbW$
\par to learn about sheaves!
\msk
\par In ``Internal Diagrams in Category Theory'' (2010)
\par I ``defined'' (loosely) a way of thinking
\par diagrammatically, and a notion of how much
\par ``mental space'' each idea takes.
\msk
\par Specializations behave like projections,
\par Generalizations behave like liftings:
\msk
%L defarrows "<-- <- <-| <."
%D diagram spec-and-gen
%D 2Dx 100 +70
%D 2D 100 G T
%D 2D
%D 2D +50 P S
%D 2D
%D (( G .tex= \arr{general\\theory}
%D P .tex= \arr{particular\\case}
%D T .tex= \arr{Toposes}
%D S .tex= \arr{`$\Set^\bbW$'s}
%D G P -> sl_ .plabel= l \arr{specialization\\(projection)}
%D G P <-- sl^ .plabel= r \arr{generalization\\(lifting)}
%D T S -> sl_
%D T S <-- sl^
%D ))
%D enddiagram
%D
$\def\arr#1{\begin{array}{c}#1\end{array}}
\def\arr#1{\begin{tabular}{c}#1\end{tabular}}
\diag{spec-and-gen}
$
\msk
\par Can we learn/define/understand sheaves
\par in toposes of the form $\Set^\bbW$
\par and then lift the theory to the general case?
% --------------------
% «zsets» (to ".zsets")
% (s "Well-positioned subsets of $Z^2$ (and ZSets)" "zsets")
\myslide {Well-positioned subsets of $Z^2$ (and ZSets)} {zsets}
\msk
\par Def: a subset $D = \{(x_1,y_1),...,(x_n,y_n)\} \subset \Z^2$
\par is {\bf well-positioned} when $\inf_i x_i = 0$ and $\inf_i y_i = 0$.
\msk
\par Def: {\bf ZSet} is a finite well-positioned subset of $\Z^2$.
\msk
\par Examples:
\par $Y = \{(0,2),(2,2),(1,1),(1,0)\}$
\par $K = \{(1,3),\,(0,2),(2,2),\,(1,1),\,(1,0)\}$
\msk
\par They will usually be named according to their shapes
\par (`K' is for `Kite').
% --------------------
% «black-pawns-moves» (to ".black-pawns-moves")
% (s "Black pawn's moves (and ZDags)" "black-pawns-moves")
\myslide {Black pawn's moves (and ZDags)} {black-pawns-moves}
\bsk
%D diagram Kite
%D 2Dx 100 +20 +20
%D 2D 100 K1
%D 2D / \
%D 2D +20 K2 K3
%D 2D \ /
%D 2D +20 K4
%D 2D |
%D 2D +20 K5
%D 2D
%D (( K1 .tex= (1,3)
%D K2 .tex= (0,2)
%D K3 .tex= (2,2)
%D K4 .tex= (1,1)
%D K5 .tex= (1,0)
%D K1 K2 -> K1 K3 ->
%D K2 K4 -> K3 K4 ->
%D K4 K5 ->
%D ))
%D enddiagram
%D
% (find-pspage "~/IMAGES/black_pawns_moves.eps")
$%
\begin{array}{c}
% \includegraphics[scale=0.35]{../IMAGES/black_pawns_moves.eps}
\includegraphics[scale=0.35]{../IMAGES/black_pawns_moves}
\end{array}
\qquad
\diag{Kite}
$
Example:
Let $K = \{(1,3),\,(0,2),(2,2),\,(1,1),\,(1,0)\}$.
Then the set of {\bf black pawn's moves} on $K$, $\BPM{K}$,
is the set of 5 arrows at the right.
Let $\bbK = (K, \BPM{K})$ \qcomment{mmm}{this a DAG.}
\bsk
Every ZSet $D$ induces a DAG
$\bbD = (D, \BPM{D})$ \qcomment{mmmmmi}{this a {\bf ZDag}.}
% --------------------
% «partial-orders» (to ".partial-orders")
% (s "Partial orders" "partial-orders")
\myslide {Partial orders} {partial-orders}
\msk
\par We are interested in S4 and categories, so
\par we like relations that are reflexive and transitive.
\par It is clumsy to draw $(Y, \BPM{Y}^*)$ (at the right),
\par so we'd like to make $(Y, \BPM{Y})$ (at the left)
\par stand for $(Y, \BPM{Y}^*)$.
%D diagram Y
%D 2Dx 100 +20 +20
%D 2D 100 Y1 Y2
%D 2D \ /
%D 2D +20 Y3
%D 2D |
%D 2D +20 Y4
%D 2D
%D (( Y1 .tex= (0,2)
%D Y2 .tex= (2,2)
%D Y3 .tex= (1,1)
%D Y4 .tex= (1,0)
%D Y1 Y3 -> Y2 Y3 -> Y3 Y4 ->
%D # Y1 loop (ul,ur)
%D # Y2 loop (ul,ur)
%D # Y3 loop (ul,ur)
%D # Y4 loop (dl,dr)
%D # Y1 Y4 -> Y2 Y4 ->
%D ))
%D enddiagram
%D
%D diagram Y*
%D 2Dx 100 +25 +25
%D 2D 100 Y1 Y2
%D 2D \ /
%D 2D +20 Y3
%D 2D |
%D 2D +20 Y4
%D 2D
%D (( Y1 .tex= (0,2)
%D Y2 .tex= (2,2)
%D Y3 .tex= (1,1)
%D Y4 .tex= (1,0)
%D Y1 Y3 -> Y2 Y3 -> Y3 Y4 ->
%D Y1 loop (ul,ur)
%D Y2 loop (ul,ur)
%D Y3 loop (ul,ur)
%D Y4 loop (dl,dr)
%D Y1 Y4 -> Y2 Y4 ->
%D ))
%D enddiagram
%D
$\diag{Y}
\two/->`<--/<400>^{\rm (saturate)}_{\rm (?)}
\diag{Y*}
$
\par Let's say that two relations, $R$ and $S$,
\par are {\bf equivalent} if $R^* = S^*$.
\par The class $[R] = \sst{S}{S^* = R^*}$ has a top element,
\par $R^*$, obtained by a kind of saturation process
\par (transitive-reflexive closure).
% --------------------
% «cycles-are-evil» (to ".cycles-are-evil")
% (s "Cycles are evil" "cycles-are-evil")
\myslide {Cycles are evil} {cycles-are-evil}
\msk
\par Let $T = (\{1,2,3\}, \{1,2,3\}^2)$ be the complete graph on $\{1,2,3\}$.
\par Then $[T]$ has two different minimal elements:
\msk
%D diagram 123
%D 2Dx 100 +15 +15
%D 2D 100 1 2
%D 2D
%D 2D +20 3
%D 2D
%D (( 1 2 -> 2 3 -> 3 1 ->
%D ))
%D enddiagram
%D
%D diagram 321
%D 2Dx 100 +15 +15
%D 2D 100 1 2
%D 2D
%D 2D +20 3
%D 2D
%D (( 1 2 <- 2 3 <- 3 1 <-
%D ))
%D enddiagram
%D
%D diagram 123*
%D 2Dx 100 +15 +15
%D 2D 100 1 2
%D 2D
%D 2D +20 3
%D 2D
%D (( 1 2 -> sl^ 1 2 <- sl_ 1 loop (ur,ul)
%D 2 3 -> sl^ 2 3 <- sl_ 2 loop (ur,ul)
%D 3 1 -> sl^ 3 1 <- sl_ 3 loop (dr,dl)
%D ))
%D enddiagram
%D
%D diagram loops-are-evil
%D 2Dx 100 +70
%D 2D 100 123
%D 2D
%D 2D +20 123*
%D 2D
%D 2D +20 321
%D 2D
%D (( 123 .tex= \diag{123} BOX
%D 321 .tex= \diag{321} BOX
%D 123* .tex= \diag{123*} BOX
%D 123 123* -> sl^ 123 123* <-- sl_
%D 321 123* -> sl^ 321 123* <-- sl_
%D ))
%D enddiagram
%D
$\diag{loops-are-evil}$
\msk
\par If we want to represent partial orders by minimal graphs
\par we will need to avoid these...
\par ``Reflexive'' arrows, i.e., those of the form $\aa \to \aa$
\par are (sort of) irrelevant, so let's ignore them:
\par Def: $R^\refl$ is $R$ plus all reflexive arrows.
\par Def: $R^\irr$ is $R$ minus all reflexive arrows.
\par Def: $R$ is acyclic when \red{$R^\irr$} has no cycles.
\qcomment{mmm}{not standard!}
\par Then in each class $[R]$ either all elements are acyclic
\par or all are cyclic.
% (find-dn4ex "edrx08.sty" "diagprep")
% (find-dn4ex "edrxdnt.tex" "defdiag")
% (find-dn5file "preamble.lua")
% (find-dn5file "README")
% (find-LATEXfile "2011ebl-slides.dnt")
% (find-LATEXfile "2010diags.dnt" "{dict-1}")
% (find-LATEXfile "2011ebl-slides.dnt" "{loops-are-evil}")
% --------------------
% «dags-are-good» (to ".dags-are-good")
% (s "DAGs are good" "dags-are-good")
\myslide {DAGs are good} {dags-are-good}
\msk
\par ``Acyclic'' for us is ``acyclic modulo reflexive arrows''...
\par Consider the set of DAGs on a finite set of vertices $A$.
\par The equivalence relation $R \sim S \iff R^*=S^*$
\par partitions it into equivalent classes that are ``diamond-shaped'',
\par i.e., ``everything between a top and a bottom element'':
\par $[R] = \sst{R'}{R^\ess \subseteq R' \subseteq R^*}$.
\par To build $R^\ess$ from $R$ we drop all ``non-essential arrows''.
\par (This is the dual of the saturation $R \mto R^*$).
\msk
\par Moral: we can represent finite partial orders canonically
\par by their minimal DAGs (that only have ``essential arrows'').
\par ZDags are finite, acyclic, and minimal. 8-)
% --------------------
% «fav-top-space» (to ".fav-top-space")
% (s "Our favorite topological space: $\bbV$" "fav-top-space")
\myslide {Our favorite topological space: $\bbV$} {fav-top-space}
\msk
%
\par Here it is:
\par as a DAG, $\bbV = (V, \BPM{V}) = (\{\aa,\bb,\cc\}, \{(\aa\to\cc),(\bb\to\cc)\})$
\par as a partial order, $\bbV = (V, \BPM{V}^*)$
\par as a top. space, $\bbV = (X, \Opens(X))$
\qcomment{mmm}{note the renaming!}
\par \lph{mm} $= (X, \{\{\aa,\bb,\cc\}, \{\aa,\cc\}, \{\bb,\cc\}, \{\cc\}, \{\}\})$
\par \lph{mm} $= (X, \{X, U, V, W, \emp\})$
\qcomment{mmmmmi}{names for the open sets}
\par \lph{mm} $= (X, \{\dagVee111, \dagVee101, \dagVee011, \dagVee001, \dagVee000\})$
\qcomment{mmmmn}{positional notation!}
\msk
\par We can think of it as a quotient topology on $\R$...
%D diagram V-as-quot
%D 2Dx 100 +15 +15 +30 +15 +15 +30 +15 +15
%D 2D 100 \aa \bb qaa qbb paa pbb
%D 2D
%D 2D +15 \cc qcc pcc
%D 2D
%D 2D +15 X QX pX
%D 2D
%D 2D +15 U V QU QV pU pV
%D 2D
%D 2D +15 W QW pW
%D 2D
%D 2D +15 \emp Qemp pemp
%D (( \aa \cc -> \bb \cc ->
%D ))
%D (( qaa .tex= (-‚,3]
%D qbb .tex= [2,+‚)
%D qcc .tex= (2,3)
%D qaa qcc -> qbb qcc ->
%D ))
%D (( paa .tex= \dagVee*¢¢{}
%D pbb .tex= \dagVee¢*¢{}
%D pcc .tex= \dagVee¢¢*{}
%D paa pcc -> pbb pcc ->
%D ))
%D (( X U -> X V -> U W -> V W -> W \emp ->
%D ))
%D (( QX .tex= (-‚,+‚)
%D QU .tex= (-‚,3)
%D QV .tex= (2,+‚)
%D QW .tex= (2,3)
%D Qemp .tex= \emp
%D QX QU -> QX QV -> QU QW -> QV QW -> QW Qemp ->
%D ))
%D (( pX .tex= \dagVee111
%D pU .tex= \dagVee101
%D pV .tex= \dagVee011
%D pW .tex= \dagVee001
%D pemp .tex= \dagVee000
%D pX pU -> pX pV -> pU pW -> pV pW -> pW pemp ->
%D ))
%D enddiagram
%D
$\diag{V-as-quot}$
\msk
\par I draw $X$ on top because it ``covers'' the other open sets,
\par and because $\dagVee111$ is $§$ (``Top'') in the Heyting algebra
\par (but $§$ is also the terminal... the HA must $\bbK^\op$).
\par Surprise: $(\Opens(X), \supseteq^\ess)$ is a ZDag!
% --------------------
% «fav-sh-and-presh» (to ".fav-sh-and-presh")
% (s "Our favorite sheaves and presheaves" "fav-sh-and-presh")
\myslide {Our favorite sheaves and presheaves} {fav-sh-and-presh}
\msk
\par Let's write $\Opens(\R)$ for $(\Opens(\R), \subseteq)$ \qcomment{mmn}{a category ($\nearrow\nwarrow$)}
\par and $\Opens(\R)^\op$ for $(\Opens(\R), \supseteq)$. \qcomment{mmmmm}{another \;\;\;\;($\swarrow\searrow$)}
\par Then $\calC^‚ Ý \Set^{\Opens(\R)^\op}$ is a \red{sheaf}.
\par Bad news: it is too big to visualize.
\msk
\par We write $\bbV \equiv \dagVee***$ and $\bbK = \bbV' \equiv \dagKite*****$.
\par Let's define presheaves $C^‚, E Ý \Set^\bbK$.
\par A {\bf} presheaf in $\Set^\bbD$ is just a functor from $\bbD$ to $\Set$.
\par \red{Sheafness} is \red{separatedness} plus \red{collatedness}.
\par $C^‚$ will obey both, and $E$ will fail both.
\par
\msk
%D diagram Cfo
%D 2Dx 100 +15 +15
%D 2D 100 \foX
%D 2D
%D 2D +15 \foU \foV
%D 2D
%D 2D +15 \foW
%D 2D
%D 2D +15 \foe
%D 2D
%D (( \foX \foU -> \foX \foV ->
%D \foU \foW -> \foV \foW ->
%D \foW \foe ->
%D ))
%D enddiagram
%D
\def\diagCfo#1#2#3#4#5{%
\def\foX{#1}%
\def\foU{#2}%
\def\foV{#3}%
\def\foW{#4}%
\def\foe{#5}%
\diag{Cfo}}
%
$ C^‚
\;\; = \;\;
\diagCfo{C^‚(X)}{C^‚(U)}{C^‚(V)}{C^‚(W)}{C^‚(\emp)}
\;\; = \;\;
\diagCfo{\calC^‚(\R)}{\calC^‚((-‚,1))\phantom{mm}}{\phantom{mm}\calC^‚((0,+‚))}{\calC^‚((0,1))}{\calC^‚(\emp)}
$
% --------------------
% «compatibility» (to ".compatibility")
% (s "Compatibility" "compatibility")
\myslide {Compatibility} {compatibility}
\msk
\par Let $U=(-‚,3)$ and $V=(2,‚)$ (temporarily).
\par Let $f_U Ý \calC^‚(U,\R)$ and $f_V Ý \calC^‚(V,\R)$, in:
\bsk
\par
$ \diagCfo{\calC^‚(\R,\R)}
{\calC^‚((-‚,3),\R)\phantom{mmmm}}
{\phantom{mmmm}\calC^‚((2,+‚),\R)}
{\calC^‚((2,3),\R)}
{\calC^‚(\emp,\R)}
$
\bsk
\par We say that two ``locally defined functions'', $f_U$ and $f_V$,
\par are {\bf compatible} iff they ``coincide wherever they're both
\par defined'' (in the example: on $(2,3)$).
\par More precisely: $f_U$ and $f_V$ are compatible iff $f_U|_{UÌV} = f_V|_{UÌV}$.
% \par (To generalize, use the morphisms: $^U_{U∧V}(f_U) = ^V_{U∧V}(f_V)$).
\par Sheafness means that every {\bf compatible family} $\{f_U, \ldots, f_V\}$
\par has exactly one glueing to an $f_{Uþ\ldotsþV}$
\par (collatedness guarantees existence of a glueing,
\par separatedness guarantees that there is at most one).
% --------------------
% «evil-presheaf» (to ".evil-presheaf")
% (s "The evil presheaf" "evil-presheaf")
\myslide {The evil presheaf} {evil-presheaf}
% \msk
\par Here is the ``evil presheaf'', $E:\dagKite***** \to \Set$.
\par Note that everything here is given explicitly ---
\par restriction functions that are the images of black pawn's moves,
\par e.g., $^X_V:E(X) \to E(V)$, are {\sl drawn};
\par restriction functions like $^U_U$ are necessarily $= \id_{E(U)}$, and
\par restriction functions like $^X_W$ are obtained by composition.
\par Note (again!) that $E$ is a {\sl functor}.
\msk
\par
%
% (find-dn4 "experimental.lua" "relphantom")
%L forths[".xtag="] = function ()
%L local x, y = ds:pick(1).x, ds:pick(1).y
%L local dx, tag = getwordasluaexpr(), getword()
%L tex = "\\ph{"..tag.."}"
%L ds[1] = storenode {x=x+dx, y=y, tex=tex, tag=tag}
%L end
%
%D diagram evil-presheaf
%D 2Dx 100 +20 +20 +30 +20 +20
%D 2D 100 A0 B0
%D 2D / \ / \
%D 2D v v v v
%D 2D +20 A1 A2 B1 B2
%D 2D \ / \ /
%D 2D v v v v
%D 2D +20 A3 B3
%D 2D | |
%D 2D v v
%D 2D +20 A4 B4
%D 2D
%D (( A0 .tex= E(X)
%D A1 .tex= E(U)
%D A2 .tex= E(V)
%D A3 .tex= E(W)
%D A4 .tex= E(\emp)
%D A0 A1 ->
%D A0 A2 ->
%D A1 A3 ->
%D A2 A3 ->
%D A3 A4 ->
%D ))
%D (( B0 .tex= \{e_X,e'_X\} place .xtag= -6 e_X .xtag= 12 e'_X
%D # B1 .tex= \{e_U,e'_U\} place .xtag= -5 e_U .xtag= 10 e'_U
%D B1 .tex= \{e_U\} place .xtag= 0 e_U .xtag= 0 e'_U
%D B2 .tex= \{e_V,e'_V\} place .xtag= -5 e_V .xtag= 11 e'_V
%D B3 .tex= \{e_W\}
%D B4 .tex= \{e_\emp\}
%D e_X e_U -> e'_X e'_U ->
%D e_X e_V -> e'_X e_V ->
%D e_U B3 -> e'_U B3 ->
%D e_V B3 -> e'_V B3 ->
%D B3 B4 ->
%D ))
%D enddiagram
%D
$ E
\;\; = \;\;
\diagCfo{E(X)}{E(U)}{E(V)}{E(W)}{E(\emp)}
\;\; = \;\;
% (find-LATEXgrep "grep -nH -e color *.tex")
\def\ph#1{{\color{red}#1}}
\def\ph#1{\phantom{#1}}
\diag{evil-presheaf}
$
\msk
\par Then $\{e_U, e_V\}$ is a compatible family,
\par because $e_U|_{UÌV} := ^U_W(e_U) = e^W$ and $e_V|_{UÌV} := ^V_W(e_V) = e^W$,
\par but $\{e_U, e_V\}$ has two different glueings, $e_X$ and $e'_X$,
\par so separatedness doesn't hold in $E$...
\par Also, $\{e_U, e'_V\}$ is another compatible family,
\par and this one has {\sl no glueings}.
\par So collatedness also doesn't hold in $E$.
% --------------------
% «stack-ops» (to ".stack-ops")
% (s "Stack operations" "stack-ops")
\myslide {Stack operations} {stack-ops}
\msk
\par The fastest way to formalize all this is by using {\bf stacks}.
\par (This is not the standard way at all! I learned it from
\par Harold Simmons's ``The point-free approach to sheafification''.)
\msk
\par This is $E$ as a stack:
\par $ÆE = E(X)÷E(U)÷E(V)÷E(W)÷E(\emp)$
\par We have an operation called ``extent'', $[e_U] = U$,
\par going from $ÆE$ to $Ø=\{X,U,V,W,\emp\}$,
\par and a non-commutative `$·$', heavily overloaded,
\par that behaves as {\sl restriction} when its left arg is in $ÆE$
\par and as {\sl intersection} when its left arg is in $Ø$:
\msk
\par
$\begin{array}{lrl}
U·V &:=& U∧V \\
&=& W \\
U·e_V &:=& U·[e_V] \\
&=& U·V \\
&=& W \\
e_U·V &:=& e_U|_{([e_U]·V)} \\
&=& e_W \\
e_U·e_V &:=& e_U|_{([e_U]·[e_V])} \\
&=& e_W \\
\end{array}
$
% --------------------
% «stack-ops-2» (to ".stack-ops-2")
% (s "Stack operations (2)" "stack-ops-2")
\myslide {Stack operations (2)} {stack-ops-2}
\msk
\par The `$·$' also accepts sets as arguments,
\par with the usual conventions:
\par $\{a,b\}·\{c,d\} = \{a·c, a·d, b·c, b·d\}$,
\par $a·\{b,c\} = \{a·b, a·c\}$,
\par $\{a,b\}·c = \{a·c, b·c\}$.
\par (Also: $[\{a,b\}] = \{[a],[b]\}$).
\msk
% --------------------
% «covers-and-families» (to ".covers-and-families")
% (s "Covers and families" "covers-and-families")
\myslide {Covers and families} {covers-and-families}
\msk
\par Def: a {\bf cover} is a subset of $Ø$. (Example: $\{U,V\}$)
\par Def: a {\bf family} is a subset of $ÆE$ ``where $[·]$ is injective''.
\par Def: a {\bf compatible family} is a family ``where `$·$' commutes''.
\par Example 1: $\{e_V,e'_V\}$ is not a family.
\par Example 2: $\{e_U,e_V\}$ is a compatible family.
\par Example 3: $\{e_X,e'_V\}$ is non-compatible family.
\msk
\par $\def\ph#1{\phantom{#1}}
\diag{evil-presheaf}
$
\msk
\par Notation for covers: $\calU, \calV, \ldots$, where $\bigcup\calV = V$.
\par Notation for families: $e_\calU$, where $[e_\calU] = \calU$.
\par Def: a cover $\calU$ is (downward) {\bf saturated} when $\calU·Ø=\calU$.
\par Def: a family $e_\calU$ is (downward) {\bf saturated} when $e_\calU·Ø=e_\calU$.
\par Example 4: $\{U,V\}·Ø = \{U,V,W,\emp\}$.
\par Example 5: $\{e_U,e'_V\}·Ø = \{e_U,e'_V,e_W,e_\emp\}$.
\par Example 6: $e_X·Ø = \{e_X, e_U, e_V, e_W, e_\emp\}$.
\par Example 7: $e_X·\{U,V\}·Ø = \{e_U, e_V, e_W, e_\emp\}$.
% --------------------
% «saturated-families» (to ".saturated-families")
% (s "Saturated families" "saturated-families")
\myslide {Saturated families} {saturated-families}
\msk
\par Let's annotate saturated covers with a `$*$'.
\par So: $\calU$, $\calU'$, $\calU^*$, $\calU^{*\prime}$ are saturated families,
\par possibly different, all ``covering $U$''.
\msk
\par Let's write the saturation operation, `$·Ø$', as `$()^*$',
\par and let's say that $\calU \approx \calV$ when $(\calU)^*=(\calV)^*$,
\par and write the equivalence classes as $[\calU]$.
\msk
\par On \red{finite DAGs} each equivalence class has both a top element
\par and a bottom element:
\par $[\calU] = \sst{\calU'}{(\calU)^¢ \subseteq \calU' \subseteq (\calU)^*}$.
\par The operation $(\calU)^¢$, that drops all ``non-essential open sets''
\par in a cover, is new...
\par and it also makes sense for families.
% \par Each \red{compatible} family can be saturated in a unique way
\msk
\par Examples:
\par $\{U,V,W\}^* = \{U,V,W,\emp\}$
\par $\{U,V,W\}^¢ = \{U,V\}$
\par $\{e_U,e_V,e_W\}^* = \{e_U,e_V,e_W,e_\emp\}$
\par $\{e_U,e_V,e_W\}^¢ = \{e_U,e_V\}$
% «.adding-unions» (to "adding-unions")
% --------------------
% «adding-unions» (to ".adding-unions")
% (s "Adding unions" "adding-unions")
\myslide {Adding unions} {adding-unions}
\msk
\par In a \red{sheaf} $F:\bbK \to \Set$ every compatible family
\par $f_\calU$ can be glued in a unique way to obtain a $f_U$,
\par and we can obtain $f_\calU$ back from $f_U$: $f_\calU = f_U·\calU$.
\msk
\par To understand what is going on here we need
\par another notion of saturation...
\msk
\par The `$*$' saturation adds {\sl smaller opens sets} to a cover;
\par The `$**$' saturation also adds {\sl unions} to a cover.
\msk
%D diagram X**
%D 2Dx 100 +30 +50 +35 +25
%D 2D 100 X ----> {X} --*--> {X}* dX dX*
%D 2D <---- <-o--- |^
%D 2D *o||**
%D 2D v|
%D 2D +30 {U,V} --*-> {U,V}* dUV dUV*
%D 2D <-o--
%D (( X .tex= X
%D {X} .tex= \{X\}
%D {X}* .tex= \{X,U,V,W,\emp\}
%D {U,V} .tex= \{U,V\}
%D {U,V}* .tex= \{U,V,W,\emp\}
%D X {X} -> sl^
%D X {X} <- sl_
%D {X} {X}* -> sl^ .plabel= a *
%D {X} {X}* <- sl_ .plabel= b ¢
%D {X}* {U,V}* --> sl_ .plabel= l *¢
%D {X}* {U,V}* <-- sl^ .plabel= r **
%D {U,V} {U,V}* -> sl^ .plabel= a *
%D {U,V} {U,V}* <- sl_ .plabel= b ¢
%D ))
%D (( dX .tex= \dagKite10000
%D dX* .tex= \dagKite11111
%D dUV .tex= \dagKite01100
%D dUV* .tex= \dagKite01111
%D dX dX* -> sl^ .plabel= a *
%D dX dX* <- sl_ .plabel= b ¢
%D dX* dUV* .> sl_ .plabel= l *¢
%D dX* dUV* <. sl^ .plabel= r **
%D dUV dUV* -> sl^ .plabel= a *
%D dUV dUV* <- sl_ .plabel= b ¢
%D ))
%D enddiagram
%D
\par $\diag{X**}$
% (find-34page 10 "Quotient topologies")
% (find-34 "" "quotient")
% (find-34page 12 "set of saturated covers")
% (find-34 "set of saturated covers")
% --------------------
% «priming» (to ".priming")
% (s "Priming" "priming")
\myslide {Priming} {priming}
\msk
\input 2011ebl-defs.tex
\def\G#1#2#3#4#5#6{\dagGuill{#1}{#2}{#3}{#4}{#5}{#6}}
\def\A{\bigGuill 123456}
\def\B{\hugeGuillPrime
{\G 111111} % 12
{(\G 101111)} % 11
{\G 011111} % 10
{\G 001111} % 8
{(\G 010111)} % 9
{\G 001011} % 7
{(\G 000111)} % 5
{(\G 001010)} % 6
{\G 000011} % 4
{(\G 000010)} % 3
{(\G 000001)} % 2
{\G 000000} % 1
}
% $$
% \bhbox{$\A$} \to \bhbox{$\B$}
% $$
%
%D diagram guill-priming
%D 2Dx 100 +70
%D 2D 100 A B
%D 2D
%D 2D +50 C D
%D 2D
%D 2D +20 E F
%D 2D
%D (( A .tex= \A
%D B .tex= \B
%D C .tex= \dagGuill******
%D D .tex= (\dagGuill******)'=\dagGuillPrime************
%D A B `-> # .plabel= a \aa|->\{\aa\}^\coint
%D C D `->
%D ))
%D enddiagram
%D
$\diag{guill-priming}$
\msk
\par To understand ``topological sheaves'' we take a DAG (e.g., $\bbV$)
\par and prime it \red{twice}; the operations `$**$' and `$*¢$' work on $\bbV''$.
\msk
\par For ``generic'' sheaves (``sheaves on a site'') we take any DAG $\bbD$
\par to play the role of $\bbV'$ and an operation `$*$' on $\bbD$
\par that obeys three rules (obeyed by `$**$', of course),
\par and from there on we treat what were ``open sets''
\par as ``truth-values'' (!!!), and the `$*$' as a modality (!!!!!).
% --------------------
% «what-next» (to ".what-next")
% (s "What next?" "what next")
\myslide {What next?} {what next}
\msk
\par ...but that doesn't fit in 20 minutes! 8-(
\par Look for the complete version of these slides in my home page!
\msk
\par \red{Goodbye! 8-)}
%:*|-*\vdash *
%:
%: ------ --------
%: P|-P^* P^*|-Q^*
%:
%:
%:
%:
%:
%:
% (find-34 "" "priming")
% (find-34page 5 "priming")
% (find-854 "" "prop-calc-in-a-hyp")
% (find-854page 88 "prop-calc-in-a-hyp")
% (find-34 "" "primingp")
% (find-34page 5 "priming")
% (find-34 "presheaves")
% (find-34page 14 "presheaves")
% (find-LATEX "2008graphs.tex")
% (find-LATEX "2011ebl-abs.tex")
%*
\end{document}
% Local Variables:
% coding: raw-text-unix
% ee-anchor-format: "«%s»"
% End: